I think it's clear that regeneration does not help when a Suffocation spell puts a Pit Fiend at 0 or -1 HP based on the following text. (Outsiders have to breathe, so Suffocation is a totally legit way to kill most of them.)
Attack forms that don’t deal hit point damage are not healed by regeneration. Regeneration also does not restore hit points lost from starvation, thirst, or suffocation.
Suffocation seems to qualify twice: it didn't deal HP "damage" (although it did drop the HP to a specified level) and it's explicitly called out as an effect that can't be healed by regeneration.
But what about Psychic Crush?
If it also fails the Fortitude save, the target is reduced to –1 hit points and is dying.
This is also not technically dealing HP damage (you go straight to a status/level), but I'm not sure how to deal with the dying condition here, since unlike Suffocation no further effect will automatically kill the target. Instead, you have to wait for it to die or stabilize.
My guess is that you should treat such a creature as having two HP tracks - one is set to -1 and starts dying (and the creature will die if HP get low enough on this track), the other tracks normal damage still subject to regeneration. I would require some outside healing to push the Psychic Crush track above -1 and negate that effect. The practical effect would be to put most (high Con bonus) regenerating creatures unconscious until they receive natural or magical healing.
Thoughts?
I would have regeneration stabilize them at -1. at which point they need magical healing to regen further; although I would probably have regeneration start functioning a gain in 24 hours if my party decided to just leave it there.
Psychic crush is not suffocation (or starvation or thirst), even if it shares a superficially similar mechanic.
Regeneration will treat it exactly as if it had taken damage equal to it's current HP+1. Because that is effectively what has happened.